Question

Answer the following question in detail.

Derive an expression for the escape speed of an object from the surface of each.

Answer 1

1. As the gravitational force due to Earth becomes zero at infinite distance, the object has to reach infinite distance in order to escape.
2. Let us consider the kinetic and potential energies of an object thrown vertically upwards with escape velocity v_e.
3. On the surface of the Earth,
   K.E. = `1/2 "mv"_"e"^2`
   P.E. = `- "GMm"/"R"`
   Total energy = P.E. + K.E.
   ∴ T.E. = `1/2"mv"_"e"^2 - "GMm"/"R"`    ....(1)
4. The kinetic energy of the object will go on decreasing with time as it is pulled back by Earth’s gravitational force. It will become zero when it reaches infinity. Thus, at infinite distance from the Earth,
   K.E. = 0
   Also, 
   P.E. = `- "GMm"/∞` = 0
   ∴ Total energy = P.E. + K.E. = 0
5. As energy is conserved
   `1/2 "mv"_"e"^2 - "GMm"/"R" = 0`     .....[From(1)]
   or, `"v"_"e" = sqrt((2"GM")/"R")`